چکیده انگلیسی

Analysis of uncertainty is often neglected in the evaluation of complex systems models, such as computational models used in hydrology or ecology. Prediction uncertainty arises from a variety of sources, such as input error, calibration accuracy, parameter sensitivity and parameter uncertainty. In this study, various computational approaches were investigated for analysing the impact of parameter uncertainty on predictions of streamflow for a water-balance hydrological model used in eastern Australia. The parameters and associated equations which had greatest impact on model output were determined by combining differential error analysis and Monte Carlo simulation with stochastic and deterministic sensitivity analysis. This integrated approach aids in the identification of insignificant or redundant parameters and provides support for further simplifications in the mathematical structure underlying the model. Parameter uncertainty was represented by a probability distribution and simulation experiments revealed that the shape (skewness) of the distribution had a significant effect on model output uncertainty. More specifically, increasing negative skewness of the parameter distribution correlated with decreasing width of the model output confidence interval (i.e. resulting in less uncertainty). For skewed distributions, characterisation of uncertainty is more accurate using the confidence interval from the cumulative distribution rather than using variance. The analytic approach also identified the key parameters and the non-linear flux equation most influential in affecting model output uncertainty.

مقدمه انگلیسی

Hydrological models have been widely used in the past to provide catchment management with information on the interaction of water, energy and vegetation processes distributed over space and time [1]. Computational models can be used to quantify surface and groundwater contributions to streamflow and salt export at catchment scale, and have particular importance with respect to the effect of changes in land-use. For example, the impact of land-usage on salt and water yield would require the evaluation of tree-planting strategies, the consequent effect on bio-diversity, and optimisation of land-use with respect to impact on stream salinity. In particular, an imbalance in the proportion of land devoted to urban, farming and forestry planning could dramatically reduce water available to streamflow and storage in a catchment.
A typical hydrological model consists of a large number of coupled equations describing the direction of water flow, including surface and sub-surface flows, providing predictions of monthly and annual streamflow or salt deposition. Additional inputs represent the spatial mosaic of climate, soil type, topography and land use [2]. Temporal inputs include estimates of surface runoff, sub-surface lateral flow, recharge, and potential evaporation. The computational procedure may estimate the partitioning between surface, lateral and groundwater pathways in the catchment and be applied over daily, monthly or yearly increments.
Historically, most hydrological models have been deterministic in that model parameters and inputs are represented by single values or point estimates — e.g. 2C or 2CSalt [3], BC2C [4] and other coupled salt and water-balance models [5]. More recently, reviews have been published on the role of calibration and uncertainty in the modelling process [1] and [6]. With respect to the specific aim of calibration and parameter estimation, uncertainty has been addressed by proposing various regression and probabilistic approaches, such as Generalised Likelihood Uncertainty Estimation (GLUE) [7], Markov Chain Monte Carlo (MCMC) [8], and also MCMC/Bayesian inference approaches, such as MCMC Global Sensitivity Analysis (MCMC-GSA) [9] and Bayesian Approach to Total Error Analysis (BATEA) [10] and [11].
In general, hydrological models incorporate many parameters (some statistical and some with physical significance), most of which require measurements from resource-intensive field exercises which are used to calibrate the model by statistical methods, such as least-squares regression analysis or the approaches cited in the previous paragraph. In some cases, parameters with physical significance may be adjusted interactively during calibration. Irrespective of the method used for parameter estimation, one often has little sense of which parameters have the most influence on model output. Indeed, some parameters may have such little impact that they could be easily ignored, leading to simplification of the mathematical structure of the model. A primary concern in this study was to explore the sensitivity of predictions to parameter variability in order to establish their relative importance for accurate calibration.
By including uncertainty in model parameters, rather than using point estimates, more information is available to the catchment manager with respect to prediction error. In a fundamental sense, uncertainty associated with model output may be represented as a probability distribution or as a specific statistical quantity, such as the 95th percentile result from the cumulative probability distribution (i.e. what is the annual streamflow prediction with a 95% probability?). By introducing notions of confidence and probability, this approach provides more information than a single point estimate and informs policy developers about the degree of risk associated with particular actions (Fig. 1).
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Fig. 1.
Uncertainty in a complex system.
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Kuczera and Parent [8] studied MCMC/Bayesian approaches to calibration uncertainty and reported a case study for the CATPRO salinity model. During calibration, they observed a bimodal histogram for one parameter, which they attributed to non-stationarity in the time-series input data. In general, however, there has been very little work reported on the shape of parameter distributions and the effect on prediction uncertainty. One objective of the current investigation was to examine the effect of parameter uncertainty on prediction uncertainty in a prototypical hydrological model (a water-balance model), where the parameters have already been assigned. The fixed parameters were subjected to systematic and random perturbations and the effect observed at the model output. The parameters were represented by distributions and the effect on prediction error and uncertainty explored by changing shape (skewness) under a variety of conditions using Monte Carlo simulation.
Another aim of the study was to present catchment managers and field hydrologists with straightforward computational strategies for the assessment of prediction uncertainty in a hydrological model. As a decision aid, this would benefit in the estimation of error, precision and confidence in the model predictions. In particular, isolation of those parameters with most effect on output would support the design and calibration of field experiments (noting cost reductions possible due to avoiding unnecessary measurement and excessive computational complexity). Whilst these investigations were carried out on the 2C hydrological model (a variant of the 2CSalt model), insights into how to deal with uncertainty are applicable to models of similar structure [3], [4] and [5]. More specific aims of the investigation were (i) to investigate the transfer of uncertainty from designated key parameters to the model output by means of appropriate metrics, (ii) to examine the influence of the shape (skewness) in the parameter distributions on prediction error, and (iii) to conduct sensitivity analysis using both point estimates and parameter distributions.

نتیجه گیری انگلیسی

Uncertainty is inherent in the structure of a predictive model, in the input data, and in the model parameters due to calibration difficulty. The advantage of uncertainty analysis is that it provides methodologies that can add value to conventional risk analysis by providing more information about the outputs of a predictive model, and identifies components of the model where uncertainties can be decreased. For an analyst, model user, or policy maker, uncertainty analysis also has the advantage of providing an error bound and confidence level on the output. This investigation did not address the issue of parameter estimation itself but rather the impact of parameter uncertainty on model output uncertainty. The main objective was to study uncertainty methodology and uncertainty propagation with hydrology as the test application. Uncertainty was explored by various methods, including differential error analysis, Monte Carlo simulation and various forms of sensitivity analysis (based on computational simplicity and suitability for shape analysis of skewed distributions). The mathematical tools used are applicable to uncertainty analysis for water-balance or similar models [3], [4] and [5] or predictive models in general.
It was clear that the model output uncertainty depended on the shape (skewness) of the parameter distributions, with increasing negative skewness leading to decreasing width of the output confidence interval (i.e. resulting in less uncertainty). Characterisation of uncertainty by variance was accurate for symmetrical distributions, but not for skewed distributions, where the preferred measure is the confidence interval from the cumulative distribution (e.g. for 90% CI, the difference between the 5th and 95th percentile values). The uncertainty analysis conducted here on the 2C hydrological model suggested that annual streamflow predictions are generally insensitive to substantial variation in the parameter values — i.e. variation greater than that likely to arise from calibration error. Sensitivity analyses identified two parameters, View the MathML sourceAQmax and β3β3, in the baseflow equation as having a dominant effect on model output uncertainty. A sensitivity series indicated that these parameters correlated negatively and positively with baseflow, respectively. Differential analysis revealed that the flux equation for baseflow, View the MathML sourceAQ, was very sensitive to β3β3 — and values of β3β3 greater than unity are associated with lower output uncertainty for View the MathML sourceAQ. The importance of this result is that View the MathML sourceAQ, or baseflow, is the dominant determinant of salt load prediction, which is a key application of the model.
The methods and findings reported here for the 2C hydrological model should prove useful for future applications with similar data inputs (noting that the parameters were perturbed in a systematic way and over a large range of values). The results are also consistent with a subjective observation by an author of a past study [2], suggesting the possible sensitivity of the model output to the parameter β3β3. The approach described for uncertainty modelling provides a means to estimate the confidence interval for the output of the 2C model, which is more valuable to the model user than a single point estimate. A more demanding exercise would be the characterisation and propagation of uncertainty arising from both parameters and data inputs, a computationally very intensive study requiring very large data sets with both temporal and spatial variation possible. This is, however, a subject for future studies.